Implicit multivariable derivative of a spherical cap

[babagoslow]

Homework Statement

Consider a spherical cap, for which the surface area and volume is

A(a,h) = \pi(a^2 +h^2)
V(a,h) = \frac{\pi h}{6}(3a^2 +h^2)

What would the aspect ratio dA/dV be?

The Attempt at a Solution

Clearly we would have

dA = 2\pi a da + 2\pi h dh
dV = \pi ha da + \frac{\pi}{2}(a^2+h^2) dh

but what would the chain rule look like? Surely, it should have more terms than just

\frac{dA}{dV } = \frac{\partial A}{\partial a} \frac{\partial a}{\partial V} + \frac{\partial A}{\partial h}\frac{\partial h}{\partial V}

, right?

 

[HallsofIvy]

Why do you think it should have more terms? a and h are the only variables.

 

[babagoslow]

If dV, dA were just functions of one variable, it would be straightforward. But here there are two variables, and perhaps I would have anticipated a chain rule of the form

\frac{dA}{dV} = \frac{\partial A}{\partial a}\left( \frac{\partial a}{\partial V} + \frac{\partial h}{\partial V} \right) + \frac{\partial V}{\partial a}\left( \frac{\partial a}{\partial A} + \frac{\partial h}{\partial A} \right)

but I'm just guessing.

 

[pasmith]

On dimensional grounds, I would expect <br /> A = V^{2/3} f\left(\frac ah\right) for some function f which can be determined from the formulae for A and V. Thus A is a function of both V and a/h.

 

[HallsofIvy]

If dV, dA were just functions of one variable, it would be straightforward. But here there are two variables, and perhaps I would have anticipated a chain rule of the form

\frac{dA}{dV} = \frac{\partial A}{\partial a}\left( \frac{\partial a}{\partial V} + \frac{\partial h}{\partial V} \right) + \frac{\partial V}{\partial a}\left( \frac{\partial a}{\partial A} + \frac{\partial h}{\partial A} \right)

but I'm just guessing.

The second part has V "in the numerator" that can't be right! "On dimensional grounds", as pasmith puts it, V has units of "length cubed", A has units of "length squared, and h and a have units of "length". \partial V/\partial a has units of "length squared", and \partial a/\partial A and \partial h/\partial A have units of "1 over length" so the second term has units of length. But dA/dV has units of "1 over length".

 

[babagoslow]

Thanks for the advice HallsofIvy and pasmith!

Pasmith, yes, you were right, I was able to evaluate the derivative by expressing the functions in terms of f(h/a). Thanks for the insight!

 

[babagoslow]

Dear Pasmith, I have been looking at this problem again and in fact I don't think it is possible to find the derivative dA/dV like you mentioned.

Let x\equiv a/h.
A^3 = V^2 f(a/h) = V^2 f(x)
3A^2 dA = 2V dV f(x) + V^2 \frac{df}{dx} dx

The complication comes in when you are trying to solve for dA/dV and you get

\frac{dA}{dV} = ... + \frac{V^2}{3A^2} \frac{df}{dx} \frac{dx}{dV}.

But note that V=\frac{\pi h}{6}(3a^2+h^2). Since the powers of h, a are different, there is no way to evaluate the derivative dx/dV without leaving behind some residual terms of a, h. The goal in the first place was to evaluate the derivative solely in x\equiv a/h. Therefore we can't parameterise the equation like you said here.

Can you help, Pasmith, or anybody? This problem has been bothering me for a while.

 

[PeroK]

A simpler approach is to use the relationship between $a$ and $h$ to express $A$ and $V$ in terms of $h$ and the radius of the sphere. That reduces it to a single variable problem.

 

[babagoslow]

A simpler approach is to use the relationship between $a$ and $h$ to express $A$ and $V$ in terms of $h$ and the radius of the sphere. That reduces it to a single variable problem.

a , h are also related by a contact angle \theta. But you can't express a spherical cap uniquely in just that one variable.

I guess I don't really understand what you meant. From what I understand a spherical cap is uniquely described by at least two parameters - one possible combination is the the base width and height a,h, and another combination is the radius of curvature and angle r,\theta.

 

[PeroK]

a , h are also related by a contact angle \theta. But you can't express a spherical cap uniquely in just that one variable.

I guess from what you said that this is not a spherical cap on a given sphere?

 

[babagoslow]

I guess from what you said that this is not a spherical cap on a given sphere?

Yes, that's right, the cap is not on a given sphere. There are two parameters.

 

[PeroK]

Yes, that's right, the cap is not on a given sphere. There are two parameters.

So, are you not trying to calculate $\frac{\partial A}{\partial V}$ ?

 

[babagoslow]

So, are you not trying to calculate $\frac{\partial A}{\partial V}$ ?

That is what I am calculating. As we know, for a sphere, V(r), A(r) and it's easy to calculate \frac{dA}{dV} = \frac{dA}{dr} \frac{dr}{dV}. But what happens when these two quantities are functions of more than one variable, ie, V(a,h), A(a,h)? Then how would we calculate \partial A/\partial V?

 

[PeroK]

That is what I am calculating. As we know, for a sphere, V(r), A(r) and it's easy to calculate \frac{dA}{dV} = \frac{dA}{dr} \frac{dr}{dV}. But what happens when these two quantities are functions of more than one variable, ie, V(a,h), A(a,h)? Then how would we calculate \partial A/\partial V?

I think your problem is that you also need to specify which variable is constant for your partial derivative. For example, you could express A as a function of V and h, say:

$A = f(V, h)$

And then you could calculate \partial A/\partial V = \partial f/\partial V where h is held constant.

But, you could also find:

$A = g(V, a)$

And then you could calculate \partial A/\partial V= \partial g/\partial V where a is held constant. And these would be different functions with different numerical values.

Or, you could hold $a/h$ constant etc.

That's why I assumed it was a fixed sphere, when you have V as a function of A with no other independent variables and can find $\frac{dA}{dV}$

 

[Ray Vickson]

Homework Statement

Consider a spherical cap, for which the surface area and volume is

A(a,h) = \pi(a^2 +h^2)
V(a,h) = \frac{\pi h}{6}(3a^2 +h^2)

What would the aspect ratio dA/dV be?

The Attempt at a Solution

Clearly we would have

dA = 2\pi a da + 2\pi h dh
dV = \pi ha da + \frac{\pi}{2}(a^2+h^2) dh

but what would the chain rule look like? Surely, it should have more terms than just

\frac{dA}{dV } = \frac{\partial A}{\partial a} \frac{\partial a}{\partial V} + \frac{\partial A}{\partial h}\frac{\partial h}{\partial V}

, right?

Your question is not well-defined (as many others have also indicated). However, it is not very difficult to extract answers to more pointed questions. We have
\Delta A = 2 \pi a \, \Delta a + 2 \pi h \, \Delta h, \\<br /> \Delta V = \pi h a \, \Delta a + \frac{\pi}{2}(a^2+h^2) \, \Delta h \\<br /> \text{so}\\<br /> \frac{\Delta A}{\Delta V} = <br /> \frac{2 \pi a \, \Delta a + 2 \pi h \, \Delta h}{\pi h a \, \Delta a + (\pi /2)(a^2+h^2) \, \Delta h}<br />
This implies the following.
(1) If $a$ is a function of $h$ we get
\frac{dA}{dV} = \frac{2 \pi a (da/dh) + 2 \pi h}{\pi h a (da/dh) + (\pi/2) (a^2 + h^2)}
(2) If $h$ is a function of $a$ we get
\frac{dA}{dV} = \frac{2 \pi a+ 2 \pi h (dh/da)}{\pi h a + (\pi/2)(a^2+h^2) (dh/da)}
(3) If $h$ is a constant, the derivative $(dA/dV)_h$ --which is Thermodynamics-style notation, indicating that $h$ is held constant--has the form
\left( \frac{d A}{d V} \right)_{h} = \frac{ 2 \pi a}{\pi h a}.
(4) If $a$ is a constant, the derivative $(dA/dV)_a$ is
\left( \frac{dA}{dV}\right)_a = \frac{2 \pi h}{(\pi/2)(a^2+h^2)}

You get many different answers, depending on what you mean by the question to begin with.